The distances from the centre of the earth, where the weight of a body is zero and one-fourth that of the weight of the body on the surface of the earth are (assume `R` is the radius of the earth)

Options

A. `0,R/4`
B. `0,(3R)/4`
C. `R/4,0`
D. `(3R)/4,0`

Explanation

To solve the problem, we need to find two distances from the center of the Earth: one where the weight of a body is zero and another where the weight is one-fourth that of the weight on the surface of the Earth.

### Step-by-Step Solution:

**Step 1: Understanding Weight Inside the Earth**
- The weight of a body at a distance \( d \) from the center of the Earth is given by the effective gravitational acceleration \( g' \) at that distance. The formula for \( g' \) at a depth \( D \) from the surface is:
\[ g' = g \left(1 - \frac{D}{R}\right) \]
where \( g \) is the gravitational acceleration at the surface and \( R \) is the radius of the Earth.

**Step 2: Finding the Distance Where Weight is Zero**
- For the weight to be zero, \( g' \) must be zero:
\[ g \left(1 - \frac{D}{R}\right) = 0 \]
This implies:
\[ 1 - \frac{D}{R} = 0 \implies D = R \]
- Therefore, at the center of the Earth (where \( D = R \)), the weight of the body is zero. The distance from the center of the Earth is:
\[ \text{Distance} = 0 \]

**Step 3: Finding the Distance Where Weight is One-Fourth**
- Now, we need to find the distance where the weight is one-fourth of the weight on the surface:
\[ g' = \frac{g}{4} \]
Substituting into the formula:
\[ g \left(1 - \frac{D}{R}\right) = \frac{g}{4} \]
Dividing both sides by \( g \):
\[ 1 - \frac{D}{R} = \frac{1}{4} \]
Rearranging gives:
\[ \frac{D}{R} = 1 - \frac{1}{4} = \frac{3}{4} \]
Thus:
\[ D = \frac{3R}{4} \]

**Step 4: Finding the Distance from the Center of the Earth**
- The depth \( D \) is measured from the surface, so the distance from the center of the Earth is:
\[ \text{Distance from center} = R - D = R - \frac{3R}{4} = \frac{R}{4} \]

### Final Answer:
The distances from the center of the Earth where the weight of a body is zero and one-fourth that of the weight on the surface of the Earth are:
\[ (0, \frac{R}{4}) \]